Ahead of time, the wizards get together and divide all B/W sequences
into classes, where two sequences are in the same class if they differ
by at most a finite number of terms. They then choose (by Axiom of
choice) one representative from each class and each wizard remembers
the representatives.
The hats are then placed on the wizards who are numbered 1, 2, 3,.....
Each wizard looks at hats higher in the sequence and identifies the
class of B/W sequences that the given sequence comes from. They then
compare the hats they see with the representative from that class.
Each guesses the corresponding color of the hat from the
representative if all the higher numbered hats agree with the
representative and the four lower number hats also agree. Otherwise
abstain. The probability of all correct will be 15/16. (the lowest
numbered wizard with all higher numbered hats agreeing with the
representative will abstain as will the next 3 wizards. The fifth
wizard in line will guess and will be correct 15/16 of the time. The
1/16 of the time he will be wrong occurs when the highest numbered
wizard disagreeing with the representative has the four hats just
below him in the sequence agreeing).